Conditions pour la bonne d´efinition des marches ` a temps continu 24

continu

La marche `a vitesse variable associ´ee au r´eseau ´electrique (G, C) peut ˆetre mal d´efinie dans des cas d´eg´en´er´es. On veut ici assurer, en particulier, qu’il n’y a pas d’explosion en temps fini, c’est-`a-dire que la marche ne saute qu’un nombre fini de fois dans un intervalle de temps fini. On rappelle des conditions permettant de montrer que (Xvar

t )_t≥0 existe et est unique en tant que processus de saut pur.

CHAPITRE 1. G´EOM´ETRIE STOCHASTIQUE ET MARCHES AU HASARD Pour cela, on introduit quelques notations permettant de voir (Xvar

t )t≥0 comme obte-nue `a partir de (Xn)n∈N par changement de temps. On se donne une famille de variables al´eatoires ind´ependantes {T<sub>n</sub>(x) : x∈S<sub>G</sub>, n∈ N} telle que les{T<sub>n</sub>(x) :n ∈N} sont de loi exponentielle de param`etre wG(x). On d´efinit alors les temps de ni`eme saut par :

R0 := 0, Rn:=Rn−1+Tn−1(Xn−1),

et la variable al´eatoire comptant le nombre de sauts de (Xvar

t )t≥0 jusqu’au temps t par :

n_∗(t) =n, siRn≤t < Rn+1.

On a ainsi Xvar

t =X_n∗(t).

Le Th´eor`eme 15.37 de [Bre68] assure que (Xvar

t )t≥0 est un processus de saut pur d`es que n_∗(t) est bien d´efini tandis que la Proposition 15.38 de ce mˆeme ouvrage en ´etablit l’unicit´e.

Cette condition est impliqu´ee par un crit`ere plus simple `a mettre en œuvre dans le cadre pr´esent pour v´erifier que (Xvar

t )t≥0 est un processus de saut pur, se trouve dans [Bre68, Proposition 15.43]. Celui-ci affirme que (X_t^var)_t≥0 est un processus de saut pur si et seulement si pour tout x∈SG :

Px "_∞ X n=0 1 wG(Xn) ^<+∞ # = 0.

Dans les Chapitres 3 et 4, on consid`erera des marches `a vitesses variables sur des graphes engendr´es par des processus ponctuels. Sous une hypoth`ese convenable d’ergodicit´e et une hypoth`ese de finitude du moment d’ordre 1 du degr´e de l’origine sous la mesure de Palm

P0 associ´ee au processus ponctuel, valides dans ces chapitres, le crit`ere pr´ec´edant peut ˆetre v´erifi´e pour P0-presque tout ξ0 en utilisant les arguments donn´es dans la preuve de [FSBS06, Proposition 10]. Ceci assure la bonne d´efinition presque sˆure des marches y apparaˆıssant.

1.2.3 L’environnement vu par la particule

Depuis les travaux de Kozlov [Koz85], Kipnis et Varadhan [KV86] et De Masi, Ferrari, Goldstein et Wick [DMFGW89], l’environnement vu par la particule est devenu un ou-til fondamental dans l’´etude des marches al´eatoires en milieux al´eatoires. Celui-ci a ´et´e abondamment utilis´e dans le contexte des marches sur Z^d associ´ees `a des conductances al´eatoires et peut ´egalement ˆetre employ´e pour ´etudier des marches sur des graphes dont l’ensemble de sommets est donn´e par une r´ealisation d’un processus ponctuel dansR^d. Les buts de cette section sont de motiver l’introduction de cette notion et de la d´efinir formel-lement dans le dernier cadre ´evoqu´e. Nous nous restreignons ici au cas du temps discret, la discussion s’adaptant sans difficult´e au cas du temps continu.

´

Etant donn´e ξ0 tir´e selon la mesure de Palm P0 d’un processus ponctuel stationnaire, consid´erons la marche simple au plus proche voisin (Xξ0

(ξ0, A(ξ0)). Notons Pξ0

x la loi de la marche partant de x. Le graphe G(ξ0) est en g´en´eral tr`es irr´egulier et il peut ˆetre avantageux de regarder les trajectoires de la marche sous la mesure dite annealed P[·] :=^R Pξ0

x [·]P0(dξ0) plutˆot que sous la loiquenched Pξ0

x . En effet, la moyennisation, due `a l’int´egration contreP0, homog´en´eise le milieu et gomme en un cer-tain sens ses irr´egularit´es. Cependant, au cours du temps, la marche gagne la connaissance des positions des points deξ0 dansR^d et des relations de voisinage entre ceux-ci ce qui en-traˆıne que (Xξ0

n )n∈Nn’est plus markovien sousP. On souhaite donc introduire un processus auxiliaire qui soit markovien sous la mesure annealed et qui permette de retrouver toute la trajectoire de (Xξ0

n )_n∈N partant de 0 ∈ ξ0. Il s’agit du processus de l’environnement vu par la particule oule marcheur (ξ0

n)n∈N d´efini par :

ξ⁰_n:=τXnξ⁰,

o`u les τ_x, x ∈ R^d, forment le groupe des translations dans R^d agissant sur N0. On note

Pξ0 la loi de ce processus ayant pour ´etat initial ξ0. Une mani`ere moins formelle de d´ecrire ce processus est d’imaginer, qu’au lieu de se d´eplacer dans ξ0, le marcheur reste statique en l’origine et enregistre les vues successives qu’il a sur l’environnement qui se d´eplace (se translate en fait) autour de lui.

Expliquons maintenant comment la marche originale (Xξ0

n )n∈N peut ˆetre reconstruite `a partir du processus de l’environnement. Pour cela, supposons que ξ0 est ap´eriodique (voir

§1.1.1) et d´efinissons ∆ sur N0× N0 par : ∆ξ⁰,ξ^e0:=

(

x si ξ^e0 =τxξ0

0 sinon ^.

Remarquons que ∆ est bien d´efini grˆace `a l’ap´eriodicit´e de ξ0; on donne une valeur ar-bitraire `a ∆ dans le cas p´eriodique. D´efinissons aussi la fonctionnelle de la trajectoire de l’environnement (ξ0 n)n∈N : Y₀ = 0 et Y_n(ξ0) := nX−1 i=0 ∆ξ0 i, ξ0 i+1 , n≥1.

Supposons que P0-presque toute r´ealisation de ξ0 soit ap´eriodique. Alors, pour presque toutξ0,Yn est bien d´efinie pour tout net la loi de Yn(ξ0)

n∈Nsous Pξ0 co¨ıncide avec P₀^ξ0, la loi de la marche originale partant de 0 dans ξ0.

Les arguments permettant d’obtenir la r´eversibilit´e et l’ergodicit´e du processus de l’en-vironnement vu par la particule par rapport `a la mesure annealed P[·] = ^R P_ξ0[·]P0(dξ0) sont relativement classiques et rappel´es dans le paragraphe 3.2.3 du Chapitre 3.

Chapitre 2

R´ecurrence et transience

Sommaire

2.1 Introduction and main results . . . . 29

2.1.1 Conditions on the point process . . . 29 2.1.2 The graph structures . . . 31 2.1.3 Conductance function . . . 31 2.1.4 Main results . . . 32 2.1.5 Outline of the paper . . . 34

2.2 A recurrence criterion . . . . 34

2.3 A transience criterion . . . . 36

2.4 Recurrence in dimension 2 . . . . 38

2.4.1 Delaunay triangulation case . . . 38 2.4.2 Skeleton of the Voronoi tiling case . . . 40

2.5 Transience in higher dimensions . . . . 43

2.5.1 Skeleton of the Voronoi tiling case . . . 43 2.5.2 Delaunay triangulation case . . . 45 2.5.3 Gabriel graph case . . . 46

2.6 Examples of point processes . . . . 51

2.6.1 Poisson point processes . . . 51 2.6.2 Mat´ern cluster processes . . . 51 2.6.3 Mat´ern hardcore processes . . . 53 2.6.4 Determinantal processes . . . 54

Dans ce chapitre, on ´etudie des marches simples ou associ´ees `a des conductances sur des graphes al´eatoires plong´es dans R<sup>d</sup>. Le but est de d´eterminer, pour presque toute r´ealisation du graphe (connexe), si ces marches sont r´ecurrentes ou transitoires. Les graphes al´eatoires que nous consid´erons d´ependent de la g´eom´etrie d’un ensemble al´eatoire, infini et localement fini de points. Plus pr´ecis´ement, ´etant donn´ee une r´ealisation ξ d’un processus ponctuel simple dans R<sup>d</sup>, un graphe G(ξ) = (V<sub>G(ξ)</sub>, E<sub>G(ξ)</sub>), connexe, infini, localement fini, est construit `a partir de la g´eom´etrie de cette r´ealisation et de r`egles d´eterministes (squelette de la mosa¨ıque de Vorono˘ı, triangulation de Delaunay, graphe de Gabriel, ...). Ce graphe est ensuite muni d’une fonction de conductance C, c’est-`a-dire une fonction strictement positive sur son ensemble d’arˆetes EG(ξ). Rappelons que la marche al´eatoire sur G(ξ) associ´ee `a C est la chaˆıne de Markov homog`ene en temps (Xn)n∈N dont les probabilit´es de transition sont donn´ees par :

P^hXn+1 =vXn =uⁱ= ^{C(u, v)}

w(u) ^, o`uw(u) := ^P_v∼uC(u, v).

Dans le Th´eor`eme 2.2, on obtient que, sous des hypoth`eses sur les variables de comp-tage du processus ponctuel sous-jacent et des hypoth`eses sur la fonction de conductance, les marches al´eatoires sur la triangulation de Delaunay, le graphe de Gabriel et le sque-lette de la mosa¨ıque de Vorono˘ı engendr´es par presque toute r´ealisation de ce processus ponctuel sont r´ecurrentes sid= 2 et transitoires si d≥3. On donne dans le Th´eoreme 2.1 des exemples de processus ponctuels satisfaisant les conditions donn´ees dans le Th´eor`eme 2.2 ; ceux-ci illustrent le fait que les r´esultats sont valables pour des processus ponctuels ayant des interactions entre points tr`es diff´erentes (attractivit´e ou r´epulsivit´e). Afin de prouver ces r´esultats, on ´etablit (et applique) deux crit`eres assez simples et esth´etiques pour la r´ecurrence ou la transience presque sˆure de marches al´eatoires. Les preuves de ceux-ci s’appuient sur une analogie bien connue entre les marches al´eatoires et les r´eseaux ´electriques. Le crit`ere de r´ecurrence (Crit`ere 2.4) est bas´e sur un d´ecoupage de l’espace en anneaux successifs et une minoration de la r´esistance efficace entre un sommet fix´e du graphe et l’infini. Notons que ce crit`ere s’applique `a des graphes d´eterministes plong´es dans Rd, d ≥ 2. L’id´ee principale de la preuve du crit`ere de transience (Crit`ere 2.8) est de d´ecouper l’espace en boˆıtes et d’injecter grossi`erement, pour presque toute r´ealisation du graphe al´eatoire, une r´ealisation d’un amas infini de percolation en r´egime sur-critique dans Zd et d’exploiter le fait qu’en dimension 3 et plus, la marche simple sur un tel amas de percolation est transitoire. Mentionnons pour terminer que ce dernier crit`ere peut ˆetre utilis´e pour obtenir la transience de marches sur lescreek-crossing graphs (voir Chapitre 3

§3.4 pour des arguments similaires).

Le travail pr´esent´e dans ce chapitre fait l’objet d’un article accept´e sous r´eserve de modifications dans Stochastic Processes and their Applications.

CHAPITRE 2. R´ECURRENCE ET TRANSIENCE

Recurrence or transience of random walks on random

graphs generated by point processes in R

^d

2.1 Introduction and main results

We deal with the question of recurrence or transience for random walks on Delaunay tri-angulations, Gabriel graphs and skeletons of Voronoi tilings which are generated by point processes in R^d. We obtain recurrence or transience results for random walks on these graphs for a.a. realization of the underlying point process. Random walks on random geometric networks are natural for describing flows, molecular diffusions, heat conduction or other problems from statistical mechanics in random and irregular media. In the last decades, most authors considered models in which the underlying graphs were the lattice

Z^d or a subgraph ofZ^d. Particular examples include random walks on percolation clusters (see [GKZ93] for the question of recurrence or transience and [BB07, Mat08, MP07] for in-variance principles) and the so-called random conductance model (see [Bis11] and references therein). In recent years, techniques were developed to analyze random walks on complete graphs generated by point processes with jump probability which is a decreasing function of the distance between points. The main results on this model include an annealed invari-ance principle [FSBS06, FM08], isoperimetric inequalities [CF09], recurrence or transience results [CFG09], a quenched invariance principle and heat kernel estimates [CFP13]. If the underlying graph is not the complete graph but, for example, a Delaunay triangulation or a Gabriel graph, additional difficulties appear due to the lack of information on the structure of the graphs (distribution and correlations of the degrees or of the edge lengths, volume growth, . . . ). Very recently, Ferrari, Grisi and Groisman [FGG12] constructed har-monic deformations of Delaunay triangulations generated by point processes which could be used to prove a quenched invariance principle via the corrector method, at least in the 2-dimensional case. Unfortunately, the sublinearity of the corrector does not directly fol-low from their proofs in higher dimensions. Precise heat kernel estimates, similar to those derived by Barlow [Bar04] in the percolation setting, are still to be obtained.

Under suitable assumptions on the point process and the transition probabilities, we obtain in this paper almost sure recurrence or transience results, namely Theorem 2.1 and Theorem 2.2, for random walks on the Delaunay triangulation, the Gabriel graph or the skeleton of the Voronoi tiling of this point process. We hope that the different methods developed in the papers [FSBS06, FM08, CF09, CFP13] can be adapted to our setting to obtain annealed and quenched invariance principles. This will be the subject of future work.

2.1.1 Conditions on the point process

In what follows the point process ξ is supposed to be simple, stationary and almost surely in general position (see [Zes08]): a.s. there are no d+ 1 points (resp. d+ 2 points) in any (d−1)-dimensional affine subspace (resp. in a sphere).

In this paper,c1, c2, . . . denote positive and finite constants. We will need the following assumptions on the void probabilities (V) and on the deviation probabilities (D2) and

(D₃+):

Assumptions.

(V) There exists a constant c1 such that for L large enough:

P^h#[0, L]^d∩ξ= 0ⁱ≤e^−c1Ld

.

(D2) If d= 2, there are constants c2, c3 such that forL, l large enough:

P^h#[0, L]×[0, l]∩ξ≥c2Llⁱ≤e^−c3Ll.

(D3+) If d≥3, there exists c4 such that for L large enough and all m >0:

P^h#[0, L]^d∩ξ≥mⁱ≤e^c4Ld−m.

For transience results, the following additional assumptions are needed:

(FRk) ξ has a finite range of dependencek, i.e., for any disjoint Borel sets A, B ⊂R^d with

d(A, B) := inf{kx−yk : x∈A, y∈B} ≥k, ξ∩A and ξ∩B are independent.

(ND) Almost surely, ξ does not have any descending chain, where a descending chain is a sequence (ui)_i∈N⊂ξ such that:

∀i∈N,kui+2−ui+1k<kui+1−uik.

As discussed in Section 2.6, these assumptions are in particular satisfied if ξ is:

• a homogeneous Poisson point process (PPP),

• a Mat´ern cluster process (MCP),

• a Mat´ern hardcore process I or II (MHP I/II).

Moreover, assumptions (V)and(D₂)hold ifξ is a stationary determinantal point process (DPP).

A brief overview on each of these point processes is given in Section 2.6. Note that these processes have different interaction properties: for PPPs there is no interaction between points, MCPs exhibit clustering effects whereas points in MHPs and DPPs repel each other.

CHAPITRE 2. R´ECURRENCE ET TRANSIENCE

2.1.2 The graph structures

Write Vorξ(x) := {x∈R^d : kx−xk ≤ kx−yk,∀y∈ξ}for the Voronoi cell of x∈ξ;x is called the nucleus or the seed of the cell. The Voronoi diagram of ξ is the collection of the Voronoi cells. It tessellates R^d into convex polyhedra. See [Møl94, Cal10] for an overview of these tessellations.

The graphs considered in the sequel are:

VS(ξ) the skeleton of the Voronoi tiling of ξ. Its vertex (resp. edge) set consists of the collection of the vertices (resp. edges) on the boundaries of the Voronoi cells. Note that this is the only graph with bounded degree considered in the sequel. Actually, if ξ is in general position in R^d, any vertex of VS(ξ) has degree d+ 1.

DT(ξ) the Delaunay triangulation of ξ. It is the dual graph of its Voronoi tiling. It has ξ

as vertex set and there is an edge between x and yin DT(ξ) if Vorξ(x) and Vorξ(y) share a (d−1)-dimensional face. Another useful characterization of DT(ξ) is the following: a simplex ∆ is a cell of DT(ξ)iff its circumscribed sphere has no point of

ξ in its interior. Note that this triangulation is well defined since ξ is assumed to be in general position.

These two graphs are widely used in many fields such as astrophysics [RBFN01], cellular biology [Pou04], ecology [Roq97] and telecommunications [BB09].

Gab(ξ) the Gabriel graph of ξ. Its vertex set is ξ and there is an edge between u, v ∈ξ if the ball of diameter [u, v] contains no point ofξin its interior. Note that Gab(ξ) is a subgraph of DT(ξ) (see [MS80b]). It contains the Euclidean minimum spanning forest ofξ (in which there is an edge between two points of ξ xandyiff there do not exist an integer m and vertices u0 = x, . . . , um =y∈ ξ such that kui −ui+1k < kx−yk

for alli∈ {0, . . . , m−1}, see [AS92]) and the relative neighborhood graph (in which there is an edge between two points x and y of ξ whenever there does not exist a third point that is closer to both x and y than they are to each other). It has for example applications in geography, routing strategies, biology and tumor growth analysis (see [BBD02, GS69, MS80a]).

2.1.3 Conductance function

Given a realization ξ of a point process and an unoriented graph G(ξ) = (V_G(ξ), E_G(ξ)) obtained fromξby one of the constructions given above, aconductanceis a positive function onEG(ξ). For any vertexusetw(u) :=^P_v∼uC(u, v) andR = 1/C the associatedresistance. Then (G(ξ), C) is an infinite electrical network and the random walk on G(ξ) associated with C is the (time homogeneous) Markov chain (Xn)n with transition probabilities given by:

P^hX_n+1 =vX_n =uⁱ= ^{C(u, v)}

We refer to [DS84, LP14] for introductions to electrical networks and random walks as-sociated with conductances. In our context, the cases where C is either constant on the edge set (simple random walk onG(ξ)) or given by a decreasing positive function of edge length are of particular interest. Our models fit into the broader context of random walks on random graphs with conductances, but due to the geometry of the problem we need to adapt the existing techniques.

2.1.4 Main results

The obtained random graph is now equipped with the conductance function C possibly depending on the graph structure. Our main result is:

Theorem 2.1. Let ξ be a homogeneous Poisson point process, a Mat´ern cluster process or a Mat´ern hardcore process of type I or II.

1. Let d= 2. Assume that, for almost any realization of ξ, C is bounded from above by a finite constant whose value possibly depends on ξ. Then for almost any realization of ξ the random walks on DT(ξ), Gab(ξ)and VS(ξ) associated with C are recurrent. 2. Let d ≥ 3. If C is uniformly bounded from below or a decreasing positive function of the edge length, then for almost any realization of ξ the random walks on DT(ξ),

Gab(ξ) and VS(ξ) associated with C are transient.

Moreover, (1) holds ifξ is a stationary determinantal point process in R².

Note that, in (1), the bounded conductances case follows immediately from the un-weighted case by Rayleigh monotonicity principle and we can restrict our attention to this last case in the proofs.

To the best of our knowledge, recurrence or transience of random walks on this kind of graphs has been sparsely considered in the literature. Only in the unpublished manuscript [ABS05] Addario-Berry and Sarkar announced similar results in the setting of simple ran-dom walks on the Delaunay triangulation generated by a PPP and they noticed that their method can be applied to more general point processes. Their proofs relied on a devia-tion result for the so-called stabbing number of DT(ξ) contained in a second unpublished manuscript (The slicing number of a Delaunay triangulation by Addario-Berry, Broutin, and Devroye). This last work is unfortunately unavailable. Note that the deviation result for thestabbing number has been proved since then in [PR12]. We develop a new method, which avoids the use of such a strong estimate and is thus more tractable. This allows us to obtain recurrence and transience results for a large class of point processes and geometric graphs.

Besides, several works show that random walks on distributional limits of finite rooted planar random graph are almost surely recurrent (see Benjamini and Schramm [BS01] and Gurel-Gurevich and Nachmias [GGN13]). Let us briefly explain how the results of [GGN13] could be used to obtain the recurrence of simple random walks on Delaunay triangulations generated by Palm measures associated with point processes in the plane. Given ξ and

CHAPITRE 2. R´ECURRENCE ET TRANSIENCE

n, consider the sub-graph Gn(ξ) of DT(ξ) with vertex set given by Vn(ξ) := {x ∈ ξ : Vorξ(x)∩[−n, n]2 6= ∅} and with edge set induced by the edges of DT(ξ). If the point process is stationary, one could obtain Delaunay triangulations generated by the Palm version of point processes rooted at 0 as distributional limits of (Gn, ρn) where, for each

n, ρn is chosen uniformly at random in Vn(ξ). Thanks to the results of Zuyev [Zuy92], we know that the degree of the origin in the Delaunay triangulation generated by the Palm measure of a PPP has an exponential tail. The proof can be adapted in the non-Poissonian case under assumptions similar to (V)and (D2) for the Palm version of the point process and an assumption of finite range of dependence. Using a similar construction, one could expect to apply the results of [BS01] to the skeleton of the Voronoi tiling VS(ξ) in the plane in which each vertex has degree 3. In this case, one must consider the Palm measure of the point process of the vertices of VS(ξ) on which there is no information. The main problem of this approach is that it provides a recurrence result for random walks on the Palm version of the point process of VS-vertices. It has no clear connection with the initial point process of the nuclei of the Voronoi cells. We do think that the recurrence criterion stated below is well adapted to the particular geometric graphs that we consider in this paper.

In the sequel, we will actually prove Theorem 2.2 which implies Theorem 2.1 and deals with general point processes as described in Subsection 2.1.1:

Theorem 2.2. Let ξ be a stationary simple point process in R^d almost surely in general position.

1. Let d = 2. Assume that ξ satisfies (V) and (D2). If C is uniformly bounded from above, then for almost any realization of ξ the random walks on DT(ξ), Gab(ξ) and

VS(ξ) associated with C are recurrent.

2. Letd≥3. Assume thatξ satisfies(V), (D3+)and(FRk). If Cis uniformly bounded from below or a decreasing positive function of the edge length, then for almost any realization ofξthe random walks onDT(ξ)andVS(ξ)associated withC are transient. If in addition ξ satisfies (ND), the same conclusion holds on Gab(ξ).

Remark 2.3. The stationarity assumption is not required. Indeed, one can derive similar results when the underlying point process is not stationary and (V), (D2) and (D₃+) are replaced by:

(V’) There exists a constant c′

1 such that for L large enough:

P^h#a+ [0, L]^d∩ξ= 0ⁱ≤e^−c′1Ld

, ∀a∈R^d.

(D’2) If d= 2, there are constants c′

2, c′

3 such that forL, l large enough:

(D’₃+) If d≥3, there exists c′

4 such that for L large enough and all m >0:

P^h#a+ [0, L]^d∩ξ≥mⁱ≤e^c′4Ld−m, ∀a∈R^d.

Let us also point out that conditions (V), (D3+) and (FRk) can be replaced by the

domination assumption (3) of Criterion 2.8 for the processes of good boxes defined in

Dans le document Marches au hasard sur des graphes géométriques aléatoires engendrés par des processus ponctuels (Page 36-157)